At the center of our treatment of modality is the idea that modal operators effect quantification over possible worlds.

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1 CHAPTER TWO SECOND STEPS: MODALITY In this chapter, we will start exploring a vast family of constructions, which are related to the attitude predicates we dealt with in the previous chapter: MODAL EXPRESSIONS. Some examples of modal constructions include these: (1) New structures must be generated. KRATZER, Angelika: Modality. In Arnim New structures can be generated. This is not absolutely impossible. This is a remote possibility. Possibly, we will return soon. Such thoughts are not expressible in any human language. This can opens at the top. This car goes twenty miles an hour. This book reads well. Bears like honey. At the center of our treatment of modality is the idea that modal operators effect quantification over possible worlds. VON STECHOW & Dieter WUNDERLICH, editors, Semantik: Ein internationales Handbuch der zeitgenössischen Forschung. Walter de Gruyter Berlin, pp KRATZER, Angelika: The Notional Category of Modality. In H. J. EIKMEYER & H. RIESER, editors, Words, Worlds, and Contexts. New Approaches in Word Semantics. de Gruyter Berlin, pp Modal Predicates as Quantifiers over Worlds Syntactic structures We will be looking here not just at the modal auxiliaries (must, may, should, can,... ), but also at infinitive-embedding main verbs and adjectives like have to, need to, be supposed to, be likely to.... Glossing over many syntactic details, the modal auxiliaries embed VPs and the main verbs/adjectives embed infinitive IPs. Adopting the VP-internal subject hypothesis 1, these two types of complements don t differ in semantic type. In fact, we will assume here that there is no semantic import to any of the additional structure that distinguishes a full infinitival clause from a bare VP. We will also assume, at least for the time being, that the main verbs and adjectives in question are 1 For the time being, the motivation for this assumption is independent of our current concerns. See, for example, the discussion of quantifier scope in ch. 8.4 of H&K. But we will take up the issue of the relative position of modals and subjects more seriously below. PAGE 2 1

2 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 all raising predicates (rather than control predicates), i.e., their subjects are not their own arguments, but have been moved from the subject-position of their infinitival complements. 2 Given all this, the structures projected by the auxiliaries and the main verbs/adjectives are essentially the same, apart from semantically vacuous material that we can ignore: (2) a. you must stay b. [ IP you [ λ 1 [ I must [ VP t 1 stay]]]] (3) a. you have to stay b. [ IP you [ λ 1 [ PRES [ t 1 λ 2 [ have [ t 2 λ 3 [ to [ t 3 stay ]]]]]]]] Actually, we will be working here with the even simpler structures below, in which the subject has been reconstructed to its lowest trace position. (E.g., these could be generated by deleting all but the lowest copy in the movement chain. 3 ) We will be able to prove that movement of a name or pronoun never affects truth-conditions, so at any rate the interpretation of the structures in (2) and (3) would be the same as of (4) and (5). As a matter of convenience, then, we will take the reconstructed structures, which allow us to abstract away from the (here irrelevant) mechanics of variable binding. (4) must [ you stay ] (5) PRES [ have-to [ you stay ]] Interpretation To make LFs like those above interpretable by our composition rules, we have to assume that the extensions of modal predicates are functions in either D t,t (truth-values to truth-values) or D st,t (propositions to truthvalues). If they were of type t, t, modal predicates would create extensional contexts. By the sort of argument we have seen above, we can show that this is not so. For example, suppose that you do in fact stay, and you also do in fact watch TV. So the truth-values (extensions) of the clauses you stay and you watch TV are the same (both are 1). Still, it may very well be true that you must stay, but false that you must watch TV. So we are left with type st, t for the modal predicate. This is, in effect, the type of generalized quantifiers over possible worlds. (Compare type et, t for quantifiers over individuals.) Indeed, an elementary intuition about the meanings of must and may is that they can be seen as expressing, respectively, a universal quantification and an existential quantification over possibilities: You must stay means (roughly) that you stay in all the acceptable (allowable, legal) possible scenarios under consideration. You may stay means that you stay in at least 2 The issue of raising vs. control will be taken up later. 3 We will talk about reconstruction in more detail later. PAGE 2 2

3 2.1 ] MODALPREDICATESASQUANTIFIERSOVERWORLDS :: SPRING2003 some of the acceptable possible scenarios (but there can be other acceptable possibilities in which you don t stay). As a first stab at the meanings of must, may, etc., then, we may write down these two (sets of) lexical entries: (6) Let C W be the set of relevant acceptable worlds. Then a. must = have-to = need-to =... = λp D s,t. w C : p(w) = 1 (in set talk: C p) b. may = can = be-allowed-to =... = λp D s,t. w C : p(w) = 1 (in set talk: p C ) We now predict (using IFA) that the LF for you must stay in (4) is true iff you stay in every world in C. And the corresponding sentence with may is predicted true iff you stay in some world in C Inferences This analysis is too crude for a couple of reasons that we will turn to presently. But it does have some desirable consequences that we will seek to preserve through all subsequent refinements. It correctly predicts a number of intuitive judgments about the logical relations between must and may and among various combinations of these items and negations. To start with some elementary facts, we feel that must φ entails may φ, but not vice versa: (7) You must stay. Therefore, you may stay. VALID (8) You may stay. Therefore, you must stay. INVALID (9) a. You may stay, but it is not the case that you must stay. 4 b. You may stay, but you don t have to stay. CONSISTENT We judge must φ incompatible with its inner negation must [not φ ], but find may φ and may [not φ ] entirely compatible: (10) You must stay, and/but also, you must leave. (leave = not stay). CONTRADICTORY (11) You may stay, but also, you may leave. CONSISTENT 4 The somewhat stilted it is not the case-construction is used in to make certain that negation takes scope over must. When modal auxiliaries and negation are together in the auxiliary complex of the same clause, their relative scope seems not to be transparently encoded in the surface order; specifically, the scope order is not reliably negation modal. (Think about examples with mustn t, can t, shouldn t, may not etc. What s going on here? This is an interesting topic which we must set aside for now.) With modal main verbs (such as have to), this complication doesn t arise; they are consistently inside the scope of clause-mate auxiliary negation. Therefore we can use (b) to (unambiguously) express the same scope order as (a), without having to resort to a biclausal structure. PAGE 2 3

4 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 We also judge that in each pair below, the (a)-sentence and the (b)-sentences say the same thing. (12) a. You must stay. b. It is not the case that you may leave. You aren t allowed to leave. (You may not leave.) 5 (You can t leave.) (13) a. You may stay. b. It is not the case that you must leave. You don t have to leave. You don t need to leave. (You needn t leave.) Given that stay and leave are each other s negations (i.e. leave = not stay, and stay = not leave ), the LF-structures of these equivalent pairs of sentences can be seen to instantiate the following schemata: 6 (14) a. must φ not [may [not φ ]] b. must [not ψ] not [may ψ] (15) a. may φ not [must [not φ ]] b. may [not ψ] not [must ψ] More linguistic data regarding the parallel logic of modals and quantifiers: HORN, Laurence: On the Semantic Properties of the Logical Operators in English. Ph.D. thesis, UCLA. Our present analysis of must, have-to,... as universal quantifiers and of may, can,... as existential quantifiers straightforwardly predicts all of the above judgments, as you can easily prove. (16) a. x φ x φ b. x ψ x ψ (17) a. x φ x φ b. x ψ x ψ Restricted Quantification over Worlds There are, however, two problems with the lexical entries in (6). First, these entries seem to be tailored to one particular reading of the modals in question, the so-called DEONTIC reading. What about other readings, such as EPISTEMIC readings, or readings that pertain to abilities, dispositions, and so on? We need a more general treatment. The second problem is that the current analysis predicts that sentences with modals are NON-CONTINGENT, that is, they are predicted to be true no matter what world they are asserted in. But that is not correct: we can 5 The parenthesized variants of the (b)-sentences are pertinent here only to the extent that we can be certain that negation scopes over the modal. In these examples, apparently it does, but as we remarked above, this cannot be taken for granted in all structures of this form. 6 In logicians jargon, must and may behave as DUALS of each other. For definitions of dual, see Barwise & Cooper (1981), p. 197; or volume 2 of Gamut (1991), p PAGE 2 4

5 2.1 ] MODALPREDICATESASQUANTIFIERSOVERWORLDS :: SPRING2003 imagine circumstances under which it is true that you must be quiet and also circumstances under which it is false. This problem can be given an especially dramatic illustration by sentences that embed one modal predicate under another: (18) It might be the case that you must leave. You might have to leave. It is compatible with what I know that the only way to attain your goal is to leave Context-Dependency Let us begin with the first problem. Traditional descriptions of modals often distinguish a number of readings: EPISTEMIC, DEONTIC, ABILITY, CIR- CUMSTANTIAL, DYNAMIC,... (Beyond epistemic and deontic, there is a great deal of terminological variety. Sometimes all non-epistemic readings are grouped together under the term ROOT.) Here are some initial illustrations. (19) EPISTEMIC MODALITY A: Where is John? B: I don t know. He may be at home. (20) DEONTIC MODALITY A: Am I allowed to stay over at Janet s house? B: No, but you may bring her here for dinner. (21) CIRCUMSTANTIAL/DYNAMIC MODALITY A: I will plant the rhododendron here. B: That s not a good idea. It can grow very tall. How are may and can interpreted in each of these examples? What do the interpretations have in common, and where do they differ? (20) is similar to the examples we considered in the last section. Applying the entry in (6), you may bring her here for dinner means that there is some world in C in which you bring her here for dinner. What is C? We characterized it as the set of acceptable or allowable worlds. What set exactly is that? Well, if (20) is a dialogue between a child and her mother, then apparently C here is intended (and understood) to be the set of those possible worlds which conform to the rules laid down by the mother (i.e., those worlds in which everyone acts in compliance with those rules). What about example (19)? Here we understand he may be at home to mean something like it is compatible with what I know that he is home. We can capture this by applying essentially the same entry (6), except that now we take C to be the set of possible worlds that conform to the speaker s knowledge (i.e., those worlds in which nothing is the case that the speaker knows is not the case). The utterance he may be at home then asserts that PAGE 2 5

6 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 there are some worlds which conform to the speaker s knowledge and in which John is at home. The example in (21) also can be seen as an existential quantification over a certain set of possible worlds. Here the set that takes the place of C in entry (6) seems to be the set of worlds which conform to the laws of nature (in particular, plant biology). What the speaker here would be saying, then, is that there are some worlds conforming to the laws of nature in which this rhododendron grows very tall. (Or is this another instance of an epistemic reading? See below for discussion of the distinction between circumstantial readings and epistemic ones.) If these analyses are on the right track, our lexical entries from (6) can pretty much stand as they are. The only refinement needed is more flexibility in the determination of the set C. Maybe this can in principle be any set of worlds whatsoever. It just needs to be salient somehow in the utterance situation in which the modalized sentence is used. The different so-called readings of the modals then are not really different readings of these lexical items, but rather result from different ways of resolving a hidden context-dependency. We encountered context-dependency before when we talked about pronouns and their referential (and E-Type) readings. We treated referential pronouns as free variables, appealing to a general principle that free variables in an LF need to be supplied with values from the utterance context. If we want to describe the context-dependency of modals in a technically analogous fashion, we can think of their LF-representations as incorporating or subcategorizing for a kind of invisible pronoun, a free variable that stands for a set of possible worlds. So we posit LF-structures like this: (22) [ I [ I must p s,t ] [ VP you quiet]] p s,t here is a variable over (characteristic functions of) sets of worlds, which like all free variables needs to receive a value from the utterance context. Possible values include: the set of worlds compatible with the speaker s current knowledge; the set of worlds in which everyone obeys all the house rules of a certain dormitory; and many others. The denotation of the modal itself now has to be of type st, st, t rather than st, t, thus more like that of a quantificational determiner rather than a complete generalized quantifier. Only after the modal has been combined with its covert restrictor do we obtain a value of type st, t. (23) a. must = have-to = need-to =... = λp D s,t. λq D s,t. w W [p(w) = 1 q(w) = 1] (in set talk: p q). b. may = can = be-allowed-to =... = λp D s,t. λq D s,t. w W [p(w) = 1 & q(w) = 1] (in set talk: p q ). PAGE 2 6

7 2.1 ] MODALPREDICATESASQUANTIFIERSOVERWORLDS :: SPRING2003 On this approach, the epistemic, deontic, etc. readings of individual occurrences of modal verbs come about by a combination of two separate things. The lexical semantics of the modal itself encodes just a quantificational force, a relation between sets of worlds. This is either the subset-relation (universal quantification; necessity) or the relation of nondisjointness (existential quantification; possibility). The covert variable next to the modal picks up a contextually salient set of worlds, and this functions as the quantifier s restrictor. The labels epistemic, deontic, circumstantial etc. group together certain conceptually natural classes of possible values for this covert restrictor. Notice that, strictly speaking, there is not just one deontic reading (for example), but many. A speaker who utters (24) You have to be quiet. might mean: I want you to be quiet, (i.e., you are quiet in all those worlds that conform to my preferences). Or she might mean: unless you are quiet, you won t succeed in what you are trying to do, (i.e., you are quiet in all those worlds in which you succeed at your current task). Or she might mean: the house rules of this dormitory here demand that you be quiet, (i.e., you are quiet in all those worlds in which the house rules aren t violated). And so on. So the label deontic appears to cover a whole openended set of imaginable readings, and which one is intended and understood on a particular utterance occasion may depend on all sorts of things in the interlocutors previous conversation and tacit shared assumptions. (And the same goes for the other traditional labels.) Epistemic vs. Circumstantial Modality: Kratzer s Example Consider sentences (25) and (26): (25) Hydrangeas can grow here. (26) There might be hydrangeas growing here. The two sentences differ in meaning in a way which is illustrated by the following scenario. Hydrangeas Suppose I acquire a piece of land in a far away country and discover that soil and climate are very much like at home, where hydrangeas prosper everywhere. Since hydrangeas are my favorite plants, I wonder whether they would grow in this place and inquire about it. The answer is (25). In such a situation, the proposition expressed by (25) is true. It is true regardless of whether it is or isn t likely that there are already hydrangeas in the country we are considering. All that matters is climate, soil, the special properties of hydrangeas, and the like. Suppose now that the country we are in has never had any con- Quoted from: KRATZER, Angelika: Modality. In Arnim VON STECHOW & Dieter WUNDERLICH, editors, Semantik: Ein internationales Handbuch der zeitgenössischen Forschung. Walter de Gruyter Berlin, pp In Kratzer (1981), the hydrangeas were Zwetschgenbäume plum trees.the German word Zwetschge, by the way, is etymologically derived from the city Damascus (Syria), the center of the ancient plum trade. PAGE 2 7

8 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 tacts whatsoever with Asia or America, and the vegetation is altogether different from ours. Given this evidence, my utterance of (26) would express a false proposition. What counts here is the complete evidence available. And this evidence is not compatible with the existence of hydrangeas. (25) together with our scenario illustrates the pure CIRCUM- STANTIAL reading of the modal can. [... ]. (26) together with our scenario illustrates the epistemic reading of modals. [... ] circumstantial and epistemic conversational backgrounds involve different kinds of facts. In using an epistemic modal, we are interested in what else may or must be the case in our world given all the evidence available. Using a circumstantial modal, we are interested in the necessities implied by or the possibilities opened up by certain sorts of facts. Epistemic modality is the modality of curious people like historians, detectives, and futurologists. Circumstantial modality is the modality of rational agents like gardeners, architects, and engineers. A historian asks what might have been the case, given all the available facts. An engineer asks what can be done given certain relevant facts. EXERCISE 2.1: Come up with examples of epistemic, deontic, and circumstantial uses of the necessity verb have to. Describe the set of worlds that constitutestheunderstood restrictorineachofyourexamples Contingency The above amendment addresses the ambiguity problem, but we still have the problem of non-contingency. Let s take a closer look at how this problem arises. When we interpret a tree like (22), we first use FA to obtain must (g c (p)) 7, and then IFA to obtain must (g c (p))(λw. you quiet w ). It is apparent that this expression picks out a unique truthvalue, independently of the evaluation world that we have chosen for the whole sentence. To pinpoint the intuitive source of the problem, let us concentrate on a particular deontic reading of (22), say, you must be quiet to conform to the house rules. Why do we feel that this is a contingent assertion? Well, the house rules can be different from one world to the next, and so we might be unsure or mistaken about what they are. In one possible world, they say that all noise must stop at 11pm, in another world they say that all noise must stop at 10pm. Suppose we know that it is 10:30 now, and that the dorm we are in has either one or the other of these two rules, but we have forgotten which. Then, for all we know, you must be quiet may be true or it may be false. Why does our current analysis fail to capture this contingency? The problem, it turns out, is with the idea that the utterance context supplies 7 g c is the contextually supplied variable assignment, and g c (p) is thus the contextually salient set of worlds over which the modal quantifies. PAGE 2 8

9 2.1 ] MODALPREDICATESASQUANTIFIERSOVERWORLDS :: SPRING2003 a determinate set of worlds as the restrictor. When I understand that you meant your use of must to quantify over the set of worlds in which the house rules of our dorm are obeyed, this does not imply that you and I have to know or agree on which set exactly this is. That depends on what the house rules in our world actually happen to say, and this may be an open question at the current stage of our conversation. What we do agree on, if I have understood your use of must in the way that you intended it, is just that it quantifies over whatever set it may be that the house rules pick out. The technical implementation of this insight requires that we think of the context s contribution not as a set of worlds, but rather as a function which for each world it applies to picks out such a set. For example, it may be the function which, for any world w, yields the set {w : the house rules that are in force in w are obeyed in w }. If we apply this function to a world w 1, in which the house rules read no noise after 10 pm, it will yield a set of worlds in which nobody makes noise after 10 pm. If we apply the same function to a world w 2, in which the house rules read no noise after 11 pm, it will yield a set of worlds in which nobody makes noise after 11 pm. Suppose, then, that the covert restrictor of a modal predicate denotes such a function, i.e., its value is of type s, st. (27) [ I [ I must R s,st ] [ VP you quiet]] And the new lexical entries for must and may that will fit this new structure are these: (28) For any w W: a. must w = have-to w = need-to w =... = λr D s,st. λq D s,t. w W [R(w)(w ) = 1 q(w ) = 1] (in set talk: (R(w) q) b. may w = can w = be-allowed-to w =... = λr D s,st. λq D s,t. w W [R(w)(w ) = 1 & q(w ) = 1] (in set talk: (R(w) q ) Let us see now how this solves the contingency problem. Notice that, in contrast to the two previous sets of lexical entries, the ones in (28) show a world-superscript on the semantic-value brackets around the modal. In other words, we now acknowledge that the modal s extension varies from one evaluation world to the next. This, as we see in the calculation below, is the key to predicting contingency for the modalized sentence as a whole. (29) Let w be a world, and let c be a context such that g c (R) = λw. λw. the house rules in force in w are obeyed in w must R you quiet w,g c = (IFA) must R w,g c (λw you quiet w ) = (FA) PAGE 2 9

10 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 must w ( R g c (λw you quiet w ) = (lex. entries you, quiet) must w ( R g c (λw. you are quiet in w ) = (lex. entry must) w W : R g c (w)(w ) = 1 you are quiet in w = (def. of g c) w W [the house rules in force in w are obeyed in w you are quiet in w ] As we see in the last line of (29), the truth-value of (27) depends on the evaluation world w. EXERCISE 2.2: Describe two worlds w 1 and w 2 so that must Ryouquiet w 1,g c =1and must Ryouquiet w 2,g c =0. EXERCISE 2.3: In analogy to the deontic relation g c (R) defined in (29), define an appropriate relation that yields an epistemic reading for a sentence likeyoumaybequiet. Are EPISTEMIC modals also contingent and embeddable? IATRIDOU, Sabine: The Past, the Possible, and the Evident. Linguistic Inquiry 21: VON FINTEL, Kai & IATRIDOU, Sabine: Epistemic Containment. URL edu/fintel/www/ec.pdf. Ms, MIT, to appear in Linguistic Inquiry Contingency and Iteration Recall our earlier example in: (30) You might have to leave. It is time to show that we now know how to deal with it. This sentence should be true in a world w iff some world w compatible with what the speaker knows in w is such that every world w in which you attain the goals that you have in w is such that you leave in w. Assume the following LF: (31) [ I [ might R 1 ] [ VP [ have-to R 2 ] [ IP you leave]]] Suppose w is the world for which we calculate the truth-value of the whole sentence, and the context maps R 1 to the function which maps w to the set of all those worlds compatible with what is known in w. might says that some of those worlds are worlds w that make the tree below might true. Now assume further that the context maps R 2 to the function which assigns to any such world w the set of all those worlds in which you attain the goals you have in w. have to says that all of those worlds are worlds w in which you leave. In other words, while it is not known to be the case that you have to leave, for all the speaker knows it might be the case. EXERCISE 2.4: Describe values for the covert s, st -variable that are intuitively suitable for the interpretation of the modals in the following sentences: (32) As far as John s preferences are concerned, you may stay with us. (33) According to the guidelines of the graduate school, every PhD candidate must take 9 credit hours outside his/her department. (34) John can run a mile in 5 minutes. PAGE 2 10

11 2.1 ] MODALPREDICATESASQUANTIFIERSOVERWORLDS :: SPRING2003 (35) This has to be the White House. (36) This elevator can carry up to 3000 pounds. For some of the sentences, different interpretations are conceivable depending on the circumstances in which they are uttered. You may therefore have to sketch the utterance context you have in mind before describing theaccessibilityrelation. EXERCISE 2.5: Collect two naturally occurring examples of modalized sentences (e.g., sentences that you overhear in conversation, or read in a newspaper or novel not ones that are being used as examples in a linguistics or philosophy paper!), and give definitions of values for the covert s, st - variable which account for the way in which you actually understood these sentences when you encountered them. (If the appropriate interpretation is not salient for the sentence out of context, include information about therelevantprecedingtextornon-linguistic background.) A technical variant of the analysis*** In our account of the contingency of modalized sentences, we adopted lexical entries for the modals that gave them world-dependent extensions of type s, st, st, t : (37) (repeated from earlier): For any w W: must w λr D s,st. λq D s,t. w W [R(w)(w ) = 1 q(w ) = 1] (in set talk: λr s,st. λq s,t. (R(w) q)). Unfortunately, this treatment somewhat obscures the parallel between the modals and the quantificational determiners, which have world-independent extensions of type et, et, t. Let s explore an alternative solution to the contingency problem, which will allow us to stick with the world-independent type- st, st, t -extensions that we assumed for the modals at first: (38) (repeated from even earlier): must = λp D s,t. λq D s,t w W [p(w) = 1 q(w) = 1](p q) (in set talk: λp D s,t. λq D s,t. p q). We posit the following LF-representation: (39) [ I [ I must [ R 4, s,st w*]] [ VP you quiet]] What is new here is that the covert restrictor is complex. The first part, R 4, s,st, is (as before) a free variable of type s, st, which gets assigned an accessibility relation by the context of utterance. The second part is a PAGE 2 11

12 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 David Dowty introduced an analogous symbol to pick out the evaluation time.we have chosen the star-notation to allude to this precedent. DOWTY, David: Tenses, Time Adverbs, and Compositional Semantic Theory. Linguistics and Philosophy 5: special terminal symbol which is interpreted as picking out the evaluation world: (40) For any w W : w* w = w. When R 4, s,st and w* combine (by Functional Application), we obtain a constituent whose extension is of type s, t (a proposition or set of worlds). This is the same type as the extension of the free variable C in the previous proposal, hence suitable to combine with the old entry for must (by FA). However, while the extension of C was completely fixed by the variable assignment, and did not vary with the evaluation world, the new complex constituent s extension depends on both the assignment and the world: (41) For any w W and any assignment a: R 4, s,st (w*) w,a = a( 4, s, st )(w). As a consequence of this, the extensions of the higher nodes I and I will also vary with the evaluation world, and this is how we capture the fact that (39) is contingent. Maybe this variant is more appealing. But in the text below, we continue to assume the original analysis as presented earlier. 2.2 Accessibility Relations and Related Concepts CARNAP, Rudolf: Meaning and Necessity. Chicago University Press Chicago. KANGER, Stig: The Morning Star Paradox. Theoria 23: HINTIKKA, Jaako: Modality and Quantification. Theoria 27: KRIPKE, Saul: A Completeness Theorem in Modal Logic. Journal of Symbolic Logic 24: KRIPKE, Saul: Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 9: For a thorough historical study, see: COPELAND, B. Jack: The Genesis of Possible Worlds Semantics. Journal of Philosophical Logic 31(2): The quantificational analysis of modal operators that we have presented so far was first developed around the 1950s in work by the philosophers Carnap, Kanger, Hintikka, and Kripke, and others. The free variables that we have introduced as the modals restrictors in our LFs are variables of type s, s, t. So their possible values are essentially BINARY RELATIONS BETWEEN POSSIBLE WORLDS (abstracting away from schönfinkelization). In the tradition of philosophical logic, these are known as ACCESSIBILITY RELATIONS. There are various aspects of the accessibility relations restricting modal operators that we can study, in particular (i) formal properties of the relations and how these affect the validity of inference patterns involving modalized sentences, and (ii) the manner in which context supplies clues as to what relation the speaker intends Reflexivity Recall that a relation is reflexive iff for any object in the domain of the relation we know that the relation holds between that object and itself. Which accessibility relations are reflexive? Take an epistemic relation: (42) R 1 := λw.λw. w is compatible with what S knows in w. We are asking whether for any given possible world w, we know that R 1 holds between w and w itself. It will hold if w is a world that is compatible with what we know in w. And clearly that must be so. Take our body of PAGE 2 12

13 2.2 ] ACCESSIBILITYRELATIONSANDRELATEDCONCEPTS :: SPRING2003 knowledge in w. The concept of knowledge crucially contains the concept of truth: what we know must be true. So if in w we know that something is the case then it must be the case in w. So, w must be compatible with all we know in w. R 1 is reflexive. We can therefore infer: [ (43) w p must w( R 1 (p) ) ] = 1 p(w) = 1 For any world w and any proposition p, if in w p must be true (relative to R 1 ), then in w p is true. If we consider a relation R 2 defined as giving us for any world w those worlds w which are compatible with what we BELIEVE in w, we no longer have reflexivity. Similarly, reflexivity is not a property of deontic accessibility relations, such as R 3 : (44) R 3 := λw. λw. w is a world in which you attain the goals that you have in w. Unfortunately, for a given world w, w is not automatically a world in which you attain the goals that you have in w. 8 EXERCISE 2.6: What about circumstantial accessibility relations? Are they reflexive? (ConsiderKratzer s hydrangeas examplefromabove.) Symmetry What would the consequences be if the accessibility relation were symmetric? Symmetry of the accessibility relation R implies the validity of the following principle: (45) Brouwer s Axiom: p w : w p [ w [ w R(w) w [w R(w ) & w p] ]] Here s the reasoning: Suppose p is true in w. Pick some arbitrary accessible world w, i.e. w R(w). Since R is assumed to be symmetric, we then have w R(w ) as well. By assumption, p is true in w, and since w is accessible from w, this means that p is true in a world accessible from w. In other words, w [w R(w ) & w p]. Since p and w were arbitrary, and w was an arbitrary world accessible from w, this establishes (45). To see whether a particular kind of modality is based on a symmetric accessibility relation, we can ask whether Brouwer s Axiom is intuitively valid with respect to this modality. If it is not valid, this shows that the accessibility relation can t be symmetric. 8 The property of reflexivity of accessibility relations can also be related to the notion of CENTERING, defined by Lewis in his work on conditionals. From a given world w, the set of accessible worlds λw. R(w)(w ) is CENTERED iff it contains w. This is equivalent to the condition that R be reflexive. PAGE 2 13

14 24.973:: SPRING2003 SECOND STEPS: MODALITY [ CHAPTER 2 HUGHES, George & CRESWELL, Max: A New Introduction to Modal Logic. Routledge London. In the case of a knowledge-based accessibility relation (epistemic accessibility), one can argue in this way that symmetry does not hold: 9 The symmetry condition would imply that if something is true, then you know that it is compatible with your knowledge (Brouwer s Axiom). This will be violated by any case in which your beliefs are consistent, but mistaken. Suppose that while p is in fact true, you feel certain that it is false, and so think that you know that it is false. Since you think you know this, it is compatible with your knowledge that you know it. (Since we are assuming you are consistent, you can t both believe that you know it, and know that you do not). So it is compatible with your knowledge that you know that not p. Equivalently 10 : you don t know that you don t know that not p. Equivalently: you don t know that it s compatible with your knowledge that p. But by Brouwer s Axiom, since p is true, you would have to know that it s compatible with your knowledge that p. So if Brouwer s Axiom held, there would be a contradiction. So Brouwer s Axiom doesn t hold here, which shows that epistemic accessibility is not symmetric. Game theorists and theoretical computer scientists who traffic in logics of knowledge often assume that the accessibility relation for knowledge is an equivalence relation (reflexive, symmetric, and transitive). But this is appropriate only if one abstracts away from any error, in effect assuming that belief and knowledge coincide. Further connections between mathematical properties of accessibility relations and logical properties of various notions of necessity and possibility are studied extensively in modal logic. EXERCISE 2.7: What can you say about the mathematical properties of the accessibility relation R that you would need to assume to make the following hold? (46) w p [p(w) = 1 can w (R)(p) = 1] Once you have found the mathematical property at stake, can you think of natural examples that show can behave this way and of others where it doesn t? Apart from Kratzer (1981, 1991), cited earlier, see also: KRATZER, Angelika: What Must and Can Must and Can Mean. Linguistics and Philosophy 1: KRATZER, Angelika: Semantik der Rede: Kontexttheorie - Modalwörter - Konditionalsätze. Scriptor Königstein/Taunus Kratzer s Conversational Backgrounds Angelika Kratzer has some interesting ideas on how accessibility relations are supplied by the context. She argues that what is really floating around in a discourse is a CONVERSATIONAL BACKGROUND. Accessibility relations can be computed from conversational backgrounds (as we shall do here), 9 Thanks to Bob Stalnaker (pc to Kai von Fintel) for help with the following reasoning. 10 This and the following step rely on the duality of necessity and possibility: q is compatible with your knowledge iff you don t know that not q. PAGE 2 14

15 2.2 ] ACCESSIBILITYRELATIONSANDRELATEDCONCEPTS :: SPRING2003 or one can state the semantics of modals directly in terms of conversational backgrounds (as Kratzer does). A conversational background is the sort of thing that is identified by phrases like what the law provides, what we know, etc. Take the phrase what the law provides. What the law provides is different from one possible world to another. And what the law provides in a particular world is a set of propositions. Likewise, what we know differs from world to world. And what we know in a particular world is a set of propositions. The intension of what the law provides is then that function which assigns to every possible world the set of propositions p such that the law provides in that world that p. Of course, that doesn t mean that p holds in that world itself: the law can be broken. And the intension of what we know will be that function which assigns to every possible world the set of propositions we know in that world. Quite generally, conversational backgrounds are functions of type s, st, t, functions from worlds to (characteristic functions of) sets of propositions. Now, consider: (47) (In view of what we know,) Brown must have murdered Smith. The in view of -phrase may explicitly signal the intended conversational background. Or, if the phrase is omitted, we can just infer from other clues in the discourse that such an epistemic conversational background is intended. We will focus on the case of pure context-dependency. How do we get from a conversational background to an accessibility relation? Take the conversational background at work in (47). It will be the following: (48) λw. λp. p is one of the propositions that we know in w. This conversational background will assign to any world w the set of propositions p that in w are known by us. So we have a set of propositions. From that we can get the set of worlds in which all of the propositions in this set are true. These are the worlds that are compatible with everything we know. So, this is how we get an accessibility relation: (49) For any conversational background f of type s, st, t, we define the corresponding accessibility relation R f of type s, st as follows: R f := λw. λw. p [f(w)(p) = 1 p(w ) = 1]. In words, w is f-accessible from w iff all propositions p that are assigned by f to w are true in w. Kratzer calls those conversational backgrounds that determine the set of accessible worlds MODAL BASES. We can be sloppy and use this term for a number of interrelated concepts: (i) the conversational background (type s, st, t ), (ii) the set of propositions assigned by the conversational background to a particular world (type st, t ), PAGE 2 15

16 MIT :: SPRING 2003 SECOND STEPS: MODALITY [ CHAPTER 2 (iii) the accessibility relation (type s, st ) determined by (i), (iv) the set of worlds accessible from a particular world (type s, t ). Kratzer calls a conversational background (modal base) REALISTIC iff it assigns to any world a set of propositions that are all true in that world. The modal base what we know is realistic, the modal bases what we believe and what we want are not. EXERCISE 2.8: Show that a conversational background f is realistic iff the corresponding accessibility relation R f (definedasin(49))isreflexive. EXERCISE 2.9: Let us call an accessibility relation TRIVIAL if it makes every world accessible from every world. R is TRIVIAL iff w w : w R(w). What would the conversational background f have to be like for the accessibility relation R f tobetrivialinthissense? EXERCISE 2.10: The definition in (49) specifies, in effect, a function from D s, st,t to D s,st. It maps each function f of type s, st, t to a unique function R f of type s, st. This mapping is not one-to-one, however. Different elements of D s, st,t may be mapped to the same value in D s,st. 11 Prove this claim. I.e., give an example of two functions f and f in D s, st,t for which (49) determines R f = R f. As you have just proved, if every function of type s, st, t qualifies as a conversational background, then two different conversational backgrounds can collapse into the same accessibility relation. Conceivably, however, if we imposed further restrictions on conversational backgrounds (i.e., conditions by which only a proper subset of the functions in D s, st,t would qualify as conversational backgrounds), then the mapping between conversational backgrounds and accessibility relations might become one-to-one after all. In this light, consider the following potential restriction: 11 In this exercise, we systematically substitute sets for their characteristic functions. I.e., we pretend that D s,t is the power set of W (i.e., elements of D s,t are sets of worlds), and D st,t is the power set of D s,t (i.e., elements of D st,t are sets of sets of worlds). On these assumptions, the definition in (49) can take the following form: (i) For any conversational background f of type s, st, t, we define the corresponding accessibility relation R f of type s, st as follows: R f := λw. {w : p [p f(w) w p]}. The last line of this can be further abbreviated to: (ii) R f := λw. f(w) This formulation exploits a set-theoretic notation which we have also used in condition (50) of the second part of the exercise. It is defined as follows: (iii) If S is a set of sets, then S := {x : Y [Y S x Y]}. PAGE 2 16

17 2.2 ] ACCESSIBILITYRELATIONSANDRELATEDCONCEPTS :: SPRING2003 (50) Every conversational background f must be closed under entailment ; i.e., it must meet this condition: w. p [ f(w) p p f(w)]. (In words: if the propositions in f(w) taken together entail p, then p must itself be in f(w).) Show that this restriction would ensure thatthemappingdefinedin (49)willbeone-to-one. EXERCISE 2.11 (THE COMPOSITIONAL MEANING OF what we know): We can fairly easily show how the intension of what we know is computed to be a function from evaluation worlds to sets of propositions known to be true in the evaluation world. Assume the following tree: (51) [ CP what [ λ 23 [ VP we [ V know p 23 ]]]] The crucial question is what the type of the trace in the object position of know is. (For this question to make sense we presuppose that there is the possibility of traces of types other than e; see e.g. H&K, pp. 212f.) Make an appropriate assumption (think about what kind of object know is expecting) and compute the intension of the LF in (51). It is much harder to think about what the semantics/pragmatics of the phraseinviewofwhatweknow hastobe. We llleavethatopen. EXERCISE 2.12: Convert the intension of the LF in (51) into an accessibility relationbyapplying(49). EXERCISE 2.13: Show that the intension of (51) meets the condition of closure under entailment that was entertained as a requirement on conversationalbackgrounds inexercise2.10. PAGE 2 17